Just-infinite $C^*$-algebras

Abstract

By analogy with the well-established notions of just-infinite groups and just-infinite (abstract) algebras, we initiate a systematic study of just-infinite C˚-algebras, i.e., infinite dimensional C˚-algebras for which all proper quotients are finite dimensional. We give a classification of such C˚-algebras in terms of their primitive ideal space, that leads to a trichotomy. We show that just-infinite, residually finite dimensional C˚-algebras do exist by giving an explicit example of (the Bratteli diagram of) an AF-algebra with these properties.Further, we discuss when C˚-algebras and˚-algebras associated with a discrete group are just-infinite. If G is the Burnside-type group of intermediate growth discovered by the first-named author, which is known to be just-infinite, then its group algebra CrGs and its group C˚-algebra C˚pGq are not just-infinite. Furthermore, we show that the algebra B " πpCrGsq under the Koopman representation π of G associated with its canonical action on a binary rooted tree is just-infinite. It remains an open problem whether the residually finite dimensional C˚-algebra Cπ pGq is just-infinite.As we shall later describe just-infinite C˚-algebras in terms of their primitive ideal space, and as the interesting cases of just-infinite C˚-algebras are those that are residually finite dimensional, we review in this section the relevant background. The primitive ideal space of a C˚-algebraA C˚-algebra A is said to be primitive if it admits a faithful irreducible representation on some Hilbert space. It is said to be prime if, whenever I and J are closed two-sided ideals in A such that I X J " 0, then either I " 0, or J " 0. It is easy to see that every primitive C˚-algebra is prime, and it is a non-trivial result that the converse holds for all separable C˚-algebras; cf. [31, Proposition 4.3.6]. However, there are (complicated) examples of non-separable C˚-algebras that are prime, but not primitive, see [37].A closed two-sided ideal I in a C˚-algebra A is said to be primitive if I ‰ A and I is the kernel of an irreducible representation of A on some Hilbert space. The primitive ideal space, PrimpAq, is the set of all primitive ideals in A. A closed two-sided ideal I of A is primitive if and only if the quotient A{I is a primitive C˚-algebra. In particular, 0 P PrimpAq if and only if A is primitive. The primitive ideal space is a T 0 -space when equipped with the hull-kernel topology, which is given as follows: the closure F of a subset F Ď PrimpAq consists of all ideals I P PrimpAq which contain Ş JPF J. If A is primitive, so that 0 P PrimpAq, then t0u " PrimpAq. In the commutative case, the primitive ideal space is the usual spectrum: PrimpC 0 pXqq is homeomorphic to X, whenever X is a locally compact Hausdorff space. The following fact will be used several times in the sequel:Remark 2.1. Each finite dimensional C˚-algebra A is (isomorphic to) a direct sum of full matrix algebras,for some positive integers n, k 1 , k 2 , . . . , k n . As each matrix algebra is simple, PrimpAq can be naturally...

Free products with amalgamation over central 𝐶*-subalgebras

“…Example 2.3 (RFD just-infinite C * -algebras). In [16], Grigorchuk, Musat, and Rørdam defined a C * -algebra to be just-infinite when it is infinite dimensional and all of its proper quotients are finite dimensional. In the same paper, they demonstrate the existence of just-infinite RFD C * -algebras.…”

“…It is worth noting that no C * -algebra can be both FDI and just-infinite. Indeed, by [16,Lemma 5.4], no RFD just-infinite C * -algebra is of type I; while, on the other hand, all FDI algebras are type I.…”

Elements of -algebras Attaining their Norm in a Finite-dimensional Representation

“…Examples of RFD C * -algebras arising from groups include full group C * -algebras of amenable maximally periodic groups [4], surface groups and fundamental groups of closed hyperbolic 3-manifolds that fiber over the circle [24], and many 1-relator groups with non-trivial center [19]. Other classes of RFD C * -algebras include amalgamated products of commutative C * -algebras [20], projective C * -algebras [21], universal C * -algebras of algebraic elements [23], the soft torus C * -algebra [11], and certain just-infinite C * -algebras [17]. (This list is certainly incomplete.)…”

Just-infinite $\boldsymbol{C^*}$-algebras and Their Invariants

“…The trace simplex of any unital residually finite dimensional C * -algebra is hence realized by a just-infinite one. We determine the trace simplex of the particular residually finite dimensional AF-algebras constructed in [3], and we show that it has precisely one extremal trace of type II 1 .We give a complete description of the Bratteli diagrams corresponding to residually finite dimensional AF-algebras. We show that a modification of any such Bratteli diagram, similar to the modification that makes an arbitrary Bratteli diagram simple, will yield a just-infinite residually finite dimensional AF-algebra.…”

“…The trace simplex of any unital residually finite dimensional C * -algebra is hence realized by a just-infinite one. We determine the trace simplex of the particular residually finite dimensional AF-algebras constructed in [3], and we show that it has precisely one extremal trace of type II 1 .…”