$C^*$-algebras from planar algebras I: canonical $C^*$-algebras associated to a planar algebra

摘要: From a planar algebra, we give functorial construction to produce numerous associated $C^*$-algebras. Our main is Hilbert $C^*$-bimodule with canonical real subspace which produces Pimsner-Toeplitz, Cuntz-Pimsner, and generalized free semicircular By compressing this system, obtain various $C^*$-algebras, including Doplicher-Roberts algebras, Guionnet-Jones-Shlyakhtenko universal (Toeplitz-)Cuntz-Krieger the newly introduced graph algebras. This first article in series studying $C^*$-algebras algebra.