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... Now consider the involutive anti-automorphism 2 defined by 2 = − − then the real structure { ∈ 2 (ℂ) ∶ 2 ( ) = * } is ℍ. Now for showing that 2 (ℝ) and ℍ are not isomorphic as real C*-algebras we use Giordano's classification of real AF-algebras [18], indeed, he has obtained an invariant which is equivalent to one by Goodearl and Handelman [19]. Giordano's invariant contains 0 , 2 and 4 , therefore since 2 ( 2 (ℝ)) ≠ 2 (ℍ) we conclude that 2 (ℝ) and ℍ are two distinct real structures of the complex C*-algebra 2 (ℂ). ...

... On the other hand, there are some cases for which there exists a unique real structure. For example, the hyperfinite 1 factor has a unique real structure ( [35], [17], [18]), and also it is shown that the injective factor ∞ has a unique real structure [18]. Now consider separable nuclear purely infinite simple C*-algebras which satisfy the Universal Coefficient Theorem in both real and complex cases. ...

... On the other hand, there are some cases for which there exists a unique real structure. For example, the hyperfinite 1 factor has a unique real structure ( [35], [17], [18]), and also it is shown that the injective factor ∞ has a unique real structure [18]. Now consider separable nuclear purely infinite simple C*-algebras which satisfy the Universal Coefficient Theorem in both real and complex cases. ...

In this paper, we present a captivating construction by Grothendieck, originally formulated for algebraic varieties, and adapt it to the realm of C*-algebras. Our main objective is to investigate the conditions under which this particular class of C*-algebras possesses a nontrivial Grothendieck ring. To achieve this, we explore the existence of nontrivial characters, which significantly enriches our understanding of these algebras. In particular, we conduct a detailed study of rings of C*-algebras over $\mathbb{C}$, $\mathbb{R}$, and $\mathbb{H}$.

... The problem is to find (up to Morita equivalence) all real AF C*-algebras A such that A<8>C=R. This is closely related to the problem of finding all involutions on R [3], [10]. ...

... Let S be a subset of {r,c, h}. Following [3] and [4], we say a real AF C*-algebra A is of type S if it can be written as a limit of a direct sum of matrix algebras over the fields indicated in S (with r representing R, c representing C, and h standing for H). Thus, if A is of type r, then A can be written as a limit of direct sums of matrix algebras over R. If A is of type re, then it can be obtained using finite dimensional matrix algebras over R and C. ...

... The classification of the type rh algebras is then analogous to that of the rank 2 situation above. Generically, there are 58 Morita equivalence classes (arising essentially from the action of GL (3,2) on the pairs of complementary subspaces of a dimension 3 vector space over the 2-element field). The nongeneric situation occurs only when there is a corresponding cubic field extension associated to KQ(R). ...

A real AF C *-algebra is the norm closure of a direct limit of finite dimensional real C *-algebras (with real *-algebra maps). When we use the unadorned “AF C *-algebra”, we mean the usual complex version.
Let R be a simple AF C *-algebra such that K 0 ( R ) is free of rank 2 or 3. The problem is to find (up to Morita equivalence) all real AF C *-algebras A such that A ꕕ C ≅ R. This is closely related to the problem of finding all involutions on R [3], [10].
For example, when the rank is 2, generically there are 8 such classes. The exceptional cases arise when the ratio of the two generators in K 0 ( R ) is a quadratic (algebraic) number, and here there are 4, 5, or 8 Morita equivalence classes, the number depending largely on the behaviour of the prime 2 in the relevant algebraic number field.

... By contrast there has been little attention paid to real C * -algebras other than the real AF-algebras considered in [9], [12], [19]. The purpose of the present paper is to show, by concentrating on the very basic example of real AI-algebras, that it can be expected that there will be appropriate real counterparts to all the complex results. ...

... The next lemma is just the appropriate version of Lemma 4.2 of [14]. It uses certain standard homomorphisms betweeen finite-dimensional real C * -algebras which were defined in [18] and [9]. ...

... where F, F are either R or H and α i is the appropriate standard homomorphism, as used in the real AF situation in Lemma 2.2 of [19] and either Theorem 3.3 of [18] or Proposition 3.6 of [9]. The effect on K-theory is correct because, as in Lemma 6.6 of [20], the evaluation of an element of M q (F) ⊗ R Z at 1 2 is a split homomorphism and therefore gives an isomorphism between the K-theory sequences for M q (F) ⊗ R Z and M q (F). ...

A classification in terms of K-theory and tracial states is obtained for those real structures which are compatible with the inductive limit structure of a simple C∗-inductive limit of direct sums of algebras of continous matrix valued functions on an interval.

... We do not go further in this direction, where one of the foremost problems is to classify the real forms of a Real C*-algebra, i.e., the classification of involutive *antiautomorphisms up to conjugacy. For a general survey see [d H], and for a solution of the classification problem in the case of special W *-algebras or AF C*-algebras see [Str] and [Grd1] or [Sty1], [Grd2], [GrH], and [GdH], respectively. ...

... We do not dwell on the complete classification of real AF C*-algebras using KO-groups which has been performed by T. Giordano et al.; cf. [Grd2], [GrH], [GdH] and [Sty1]. ...

... An irreducible representation of C(X i , η i ) is a point-evaluation map. By [13,Lemma 3.5], any homomorphism (including irreducible representations) from M n i (F i ) into another real matrix algebra is unitarily equivalent to a standard homomorphism. In summary, there exist a unitary u ∈ F ⊗ R M n j (F j ), F ∈ {R, C}, a standard homomorphism µ : ...

Abstract. In this paper, a classification of simple unital real C∗-al-
gebras that are inductive limits of certain real circle algebras such as
C(T,M n2(H)) is given. The invariant consists of certain triples of real
K-groups and the tracial state space of the complexification.

... This has a rather surprising consequence: if R R is the (unique) real hyperfinite II 1 factor, then R R ⊗ R H ∼ = R R (since this is a real form of R ⊗ M 2 (C) ∼ = R). In fact we also have Theorem 1.8 ( [20]). The injective II ∞ factor has a unique real form. ...

For a long time, practitioners of the art of operator algebras always worked
over the complex numbers, and nobody paid much attention to real C*-algebras.
Over the last thirty years, that situation has changed, and it's become
apparent that real C*-algebras have a lot of extra structure not evident from
their complexifications. At the same time, interest in real C*-algebras has
been driven by a number of compelling applications, for example in the
classification of manifolds of positive scalar curvature, in representation
theory, and in the study of orientifold string theories. We will discuss a
number of interesting examples of these, and how the real Baum-Connes
conjecture plays an important role.

... Using that isomorphism, part (5) follows from part (2). Finally, part (6) follows from (5) and the formula M 2 ∞ ⊗ H ∼ = M 2 ∞ , which follows from Theorem 10.1 of [21] or from Theorem 4.8 of [42]). ...

We classify real Kirchberg algebras using united K- theory. Precisely, let A and B be real simple separable nuclear purely infinite C*-algebras that satisfy the universal coefficient theorem such that AC{black-letter} and BC{black-letter} are also simple. In the stable case, A and B are isomorphic if and only if KCRT (A) ≅ KCRT (B). In the unital case, A and B are isomorphic if and only if (KCRT (A), [1A]) ≅ (KCRT (B), [1B]). We also prove that the complexification of such a real C*-algebra is purely infinite, resolving a question left open from [43]. Thus the real C*-algebras classified here are exactly those real C*-algebras whose complexification falls under the classification result of Kirchberg [26] and Phillips [35]. As an application, we find all real forms of the complex Cuntz algebras On for 2 ≤ n ≤ ∞.

... Each involutory *-antiautomorphism of an AF algebra A has associated with it a real subalgebra R A of the AF algebra. These subalgebras are called real AF algebras and there exists also a classification of real AF algebras [12], and thus, for a fixed pair of AF algebras with given involutions, there exists a natural correspondence between the following three objects: i) homomorphisms of ordered real K-theory groups, f : ...

We construct Hopf algebra isomorphisms of discrete multiplier Hopf
C*-algebras, and Hopf AF C*-algebras (generalized quantum UHF algebras), from
K-theoretical data. Some of the intermediate results are of independent
interest, such as a result that Jordan maps of Hopf algebras intertwine
antipodes, and the applications to automorphisms of Hopf algebras.

... This raises the possibility of classifying all real structures in the algebras under consideration by developing a real analogue of Lin's classification [10] of C * -algebras of tracial rank zero. Previously all classifications of real structures in non-type I simple C * -algebras, such as [2], [3], [15] for AF algebras and [5], [16] for irrational rotation algebras, have assumed very specific forms for the real algebras. ...

Let A be a simple unital C∗-algebra with tracial rank zero and with a unique tracial state and let Φ be an involutory ∗-antiautomorphism of A. It is shown that the associated real algebra AΦ = {a ∈ A :Φ (a )= a∗} also has tracial rank zero. Let A be a unital simple separable C ∗-algebra with tracial rank zero and suppose that A has a unique tracial state. If Φ is an involutory ∗-antiautomorphism of A, then it is clear that the associated real algebra AΦ = {a ∈ A :Φ (a )= a∗} is unital and simple with a unique tracial state, but it is not clear that it has tracial rank zero, even when A is approximately finite-dimensional. The purpose of the present note is to show that techniques recently developed by Phillips (14) and Osaka and Phillips (12), (13) can be used to show that AΦ does have tracial rank zero. This raises the possibility of classifying all real structures in the algebras under consideration by developing a real analogue of Lin's classification (10)

... The * -homomorphisms between finite-dimensional R * -algebras are known, say by [6]. Up to unitary equivalence, the only embedding of C into M n (H) is found be the obvious embedding into ...

We extend Jordan's notion of principal angles to work for two subspaces of
quaternionic space, and so have a method to analyze two orthogonal projections
in M_n(A) for A the real, complex or quaternionic field (or skew field). From
this we derive an algorithm to turn almost commuting projections into commuting
projections that minimizes the sum of the displacements of the two projections.
We quickly prove what we need using the universal real C*-algebra generated by
two projections.

The Toms–Winter conjecture is verified for those separable, unital, nuclear, infinite-dimensional real C *-algebras for which the complexification has a tracial state space with compact extreme boundary of finite covering dimension.

This guided lists of literature contains a collection of research papers and preprints on the theory of Hilbert C*-modules and their applications in various spheres of mathematics.
Enjoy!

this is a guided literature list on the subject hilbert C*-modules and their applications. The state is as of April 2016. I continue to work on a follow-up edition.

A K-theoretic classification is given of those symmetries on real AF C∗-algebras that arise from inductive limits of symmetries on finite dimensional real C∗-algebras. The invariant consists of a diagram of six K0 groups, with their order structures, certain distinguished elements, a distinguished sub-semigroup of one of them, and period two automorphisms of the K0 groups. As a corollary, we get that cocycle conjugacy and equivariant isomorphism coincide for these systems.

Clifford algebras of real Hilbert C*-modules with orthonormal bases are introduced. It is showed that the C*-Clifford algebra of a real Hilbert module over a real C*-algebra is *-isomorphic to the spacial tensor product of the complexification C*-algebra and a UHF C*-algebra. The von Neumann Clifford algebra of a real Hilbert module over a von Neumann algebra is *-isomorphic to the von Neumann algebra tensor product of the complexification von Neumann algebra and the hyper-finite type II1 factor.

We obtain two characterizations of the bi-inner Hopf *-automorphisms of a finite-dimensional Hopf C*-algebra, by means of an analysis of the structure of convolution products in this class of Hopf C*-algebra.

A classification is given of certain separable nuclear C*-algebras of real rank zero and stable rank one. Theses termes refer to density of the invertible elements within the sets of selfadjoint elements and all elements, respectively, after a unit has been adjoined. The C*-algebras considered are those that can be expressed as the inductive limit of a sequence of subhomogeneous C*-algebras, each with spectrum a finite union of circles and intervals, with fibre constant over each circle, and over each interval except at the endpoints -at which it is the same. While these basic building blocks may appear rather special, their limits may turn out to be quite general. (The only obvious restriction is that KQ must be torsion free.) The invariant which effects the classification is the abelian group K* = K1 + K1, together with the distinguished subset consisting of the pairs ([e], [u]) where e is a projection in the algebra and w is a partial unitary with support (and ränge) e. The K0 part of this data is exactly that of an AF algebra. In the simple case, the only additional Information needed is the K1-group, which may be any countable abelian group as soon as the K0-group is different from 0 and Z. Embedded in the classification is the fact that any C*-algebra in the class considered with ÄK-group equal to zero is an AF algebra. It follows that for any C*-algebra in this class with torsion Kl-group, the tensor product with the universal Glimm UHF algebra (with K0 = Q and K1 = 0), which is again in this class and has zero K1-group, is an AF algebra. An example of this phenomenon was discovered recently by Evans and Kishimoto ([13]).

An irrational rotation C * -algebra can be written as a direct limit of algebras, each of which is the direct sum of two matrix algebras over the algebra of continuous functions on the circle. The invariance of such decompositions under involutory antiautomorphisms is investigated, and a complete answer obtained for toral antiautomorphisms.

A 4-cycle algebra is a finite-dimensional digraph algebra (CSL algebra) whose reduced digraph is a 4-cycle. A rigid embedding between such algebras is a direct sum of certain nondegenerate multiplicity one star-extendible embeddings. A complete classification is obtained for the regular isomorphism classes of direct systemsAof 4-cycle algebras with rigid embeddings. The critical invariant is a binary relation inK0A⊕H1A, generalising the scale of theK0group, called the joint scale. The joint scale encapsulates other invariants and compatibility conditions of regular isomorphism. These include the scale ofH1A, the scale ofH0A⊕H1A, sign compatibility, congruence compatibility andH0H1coupling classes. These invariants are also important for liftingK0⊕H1isomorphisms to algebra isomorphisms; we resolve this lifting problem for various classes of 4-cycle algebra direct systems

The behavior of fermionic systems depends on the geometry of the system and the symmetry class of the Hamiltonian and observables. Almost commuting matrices arise from band-projected position observables in such systems. One expects the mathematical behavior of almost commuting Hermitian matrices to depend on two factors. One factor will be the approximate polynomial relations satisfied by the matrices. The other factor is what algebra the matrices are in, either \({{\bf M}_n(\mathbb{A})}\) for \({\mathbb{A} = \mathbb{R}}\) , \({\mathbb{A} = \mathbb{C}}\) or \({\mathbb{A} = \mathbb{H}}\) , the algebra of quaternions.
There are potential obstructions keeping k-tuples of almost commuting operators from being close to a commuting k-tuple.We consider two-dimensional geometries and so this obstruction lives in \({KO_{-2}(\mathbb{A})}\) . This obstruction corresponds to either the Chern number or spin Chern number in physics. We show that if this obstruction is the trivial element in K-theory then the approximation by commuting matrices is possible.

This is a guided list of mathematical literature of the period 1952-now in each listet reference of which the concept of Hilbert C*-modules plays a role or, even more, the theory of Hilbert C*-modules is developed further. The list contains about 1750 items. A guide through the references concerning several selected subjects stands in the beginning. It is searchable within a PDF reader.
The original updated list can be found at www.imn.htwk-leipzig.de/~mfrank/hilmod.html. I intend to continue this survey.

We give a survey on the homotopy theory of the regular group of Banach algebras with emphasis on the unstable K-Theory of real and complex C*-algebras

We define united KK-theory for real C*-algebras A and B such that A is separable and B is sigma-unital, extending united K-theory in the sense that KK\crt(\R, B) = K\crt(B). United KK-theory contains real, complex, and self-conjugate KK-theory; but unlike unaugmented real KK-theory, it admits a universal coefficient theorem. For all separable A and B in which the complexification of A is in the bootstrap category, KK\crt(A,B) can be written as the middle term of a short exact sequence whose outer terms involve the united K-theory of A and B. As a corollary, we prove that united K-theory classifies KK-equivalence for real C*-algebras whose complexification is in the bootstrap category.

The overview contains 450 references of books, chapters of monographs, papers, preprints and Ph.~D.~thesises which are concerned with the theory and/or various applications of Hilbert C*-modules. To show a way through this amount of literature a four pages guide is added clustering sources around major research problems and research fields, and giving information on the historical background. Two smaller separate parts list references treating Hilbert modules over Hilbert*-algebras and Hilbert modules over (non-self-adjoint) operator algebras. Any additions, corrections and forthcoming information are welcome.